https://file.aiquickdraw.com/mathbot/file/1726536240968b7ceu6cw.jpg

The image depicts an angle system with intersecting lines at point \( Q \), where several angle expressions are given. Let's analyze and solve it:

### Angles Given:
1. \( \angle PQS = (5x + 16)^\circ \)
2. \( \angle PQR = (3x + 12)^\circ \)
3. \( \angle SQT = (6y - 21)^\circ \)

Since the angles \( \angle PQS \) and \( \angle PQR \) are on a straight line (a linear pair), the sum of these two angles must equal \( 180^\circ \) because they form a straight angle.

### Equation 1 (for \( x \)):
\[
(5x + 16) + (3x + 12) = 180
\]
Simplifying:
\[
8x + 28 = 180
\]
\[
8x = 152
\]
\[
x = 19
\]

### Next, solving for \( y \):
The angle \( \angle SQT \) and \( \angle PQS \) appear to be vertical angles, which are always congruent (equal).

Thus, we have:
\[
5x + 16 = 6y - 21
\]
Substituting \( x = 19 \) from the previous solution:
\[
5(19) + 16 = 6y - 21
\]
\[
95 + 16 = 6y - 21
\]
\[
111 = 6y - 21
\]
\[
6y = 132
\]
\[
y = 22
\]

### Final values:
- \( x = 19 \)
- \( y = 22 \)

Would you like further explanation or details on this solution?

Here are 5 related questions:
1. How are linear pairs of angles used to set up equations in geometry?
2. What are vertical angles, and why are they always equal?
3. Can you solve a similar problem involving different angle relationships?
4. How would the solution change if one angle were given instead of an expression?
5. What real-life applications involve solving angles?

**Tip:** Always look for linear pairs and vertical angles when dealing with intersecting lines, as these relationships simplify many problems.

Find the values of x and y in the angle system where \( \angle PQS = (5x + 16)^\circ \), \( \angle PQR = (3x + 12)^\circ \), and \( \angle SQT = (6y - 21)^\circ \).

Learn how to solve an angle system involving algebraic expressions and geometry concepts, including linear pairs and vertical angles. The problem gives algebraic angles, requiring solving for x and y using congruence and the sum of angles. Suitable for students in Grades 9-11.

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